Finite Group Databasebeta

← #329714

Subgroups — #329714

Order 512.

675 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 675 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32957C212yesmaximal
2#29876C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic
4#10626C212yesmaximal, complemented
5#10483C212yesmaximal, complemented
6#10482C212yesmaximal, complemented
7#10279C212yesmaximal, complemented
8#9136C212yesmaximal
9#8600C211yesmaximal, characteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#500C2211yescharacteristic
12C2×#477C2212yes—
13C2×#477C2212yes—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2212yes—
17#2086—22no—
18#2075—22no—
19#1990C2212yes—
20#1985—22no—
21#1984C2212yes—
22#1984C2212yes—
23#1863—22no—
24#1863—22no—
25#1852—22no—
26#1834C2211yescharacteristic
27#1817—22no—
28#1803C2212yes—
29#1803C2212yes—
30#1797C2212yes—
31#1787—22no—
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#333C2212yes—
35C2×#333C2212yes—
36#1296C2212yes—
37#1296C2212yes—
38#1277C2212yes—
39C23×Q8D411yescharacteristic
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—41no—
45C2×C4×Q8—41no—
46C2×C4×Q8—41no—
47C2×C4×Q8—41no—
48C2×C4×Q8D412yes—
49C2×C4×Q8C2312yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C42—21no—
53C22×C42C2311yescharacteristic
54#500—41no—
55#500—41no—
56#477—42no—
57#477—42no—
58#477—42no—
59#477—42no—
60#432—41no—
61#432—41no—
62#431—41no—
63#431—41no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8D412yes—
78C2×#97—21no—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#391—22no—
82#389—22no—
83#389—22no—
84#384C2312yes—
85#380—22no—
86#380—22no—
87#340—22no—
88#340—24no—
89#340—22no—
90#340—22no—
91#336—24no—
92#336—24no—
93#336—22no—
94#335—41no—
95#334—41no—
96#333—42no—
97#333—42no—
98C22×Q8—21no—
99C22×Q8—24no—
100C22×Q8—22no—
101C22×Q8—41no—
102C22×Q8—41no—
103C22×Q8—42no—
104C22×Q8C2×D412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C22⋊Q8—84no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—42no—
125C4×Q8—42no—
126C4×Q8—42no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C2×D412yes—
144C2×C22⋊C4—42no—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—42no—
152C2×C42—42no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158#107—44no—
159#107—44no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—42no—
167#103—44no—
168#103—44no—
169#103—44no—
170#103—44no—
171C22⋊C8—42no—
172C22⋊C8—44no—
173C22⋊C8—42no—
174#97—42no—
175C4×C8—21no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—21no—
179#95—42no—
180#95—42no—
181C24#12111yescharacteristic
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—42no—
189C2×Q8—44no—
190C2×Q8—48no—
191C2×Q8—44no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C2×Q8—82no—
196C2×Q8#13714yes—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—44no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C2×#4111yescharacteristic
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—44no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—82no—
249C4⋊C4—84no—
250C22⋊C4—84no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—44no—
258C42—42no—
259C42—42no—
260C42H(5,𝔽2)11yescharacteristic
261C42H(5,𝔽2)11yescharacteristic
262C42H(5,𝔽2)11yescharacteristic
263C42H(5,𝔽2)11yescharacteristic
264C23—21no—
265C23—22no—
266C23—22no—
267C23—21no—
268C23#47212yes—
269C23C2×#12111yescharacteristic
270Q8—24no—
271Q8—24no—
272Q8—48no—
273Q8—42no—
274Q8—42no—
275Q8—162no—
276Q8—162no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—48no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#53411yescharacteristic
308C2×C4#53411yescharacteristic
309C2×C4#53411yescharacteristic
310C2×C4#53511yescharacteristic
311C2×C4#53811yescharacteristic
312C2×C4#53811yescharacteristic
313C2×C4C2×#13712yes—
314C2×C4#57512yes—
315C2×C4#57814yes—
316C8—42no—
317C8—42no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—24no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22#261711yescharacteristic
327C22#304412yes—
328C22#305612yes—
329C22C2×#47211yescharacteristic
330C22#306711yescharacteristic
331C4—21no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—44no—
336C4—41no—
337C4—82no—
338C4—162no—
339C4—162no—
340C4#319511yescharacteristic
341C4#319612yes—
342C4#319711yescharacteristic
343C4#320212yes—
344C4#322312yes—
345C2—42no—
346C2#1947712yes—
347C2#1947812yes—
348C2#2972411yescharacteristic
349C2#2974311yescharacteristic
350C2#2974811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #10626C2semidirect
5. #10483C2semidirect
6. #10482C2semidirect
7. #10279C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#329572
#298762
#218202
#106262
#104832
#104822
#102792
#91362
#86002